Question

An armature-controlled dc motor is sometimes used in speed
andposition control systems. The dc motor operation is described
bythe following equations:

where:

e = armature voltage, V

i = armature current, A

ω = motor speed, rad/s

q = motor torque, N.m

J = moment of inertia of the load, kg.m2

b = damping resistance of the load, N.m/(rad/s)

R = armature resistance, Ω

L = armature inductance, H

Ke = back emf constant of the motor,V/(rad/s)

Kt = torque constant of the motor, N.m/A

A small permanent-magnet dc motor has the following
parametervalues:

J = 8 x 10-4 kg.m2

b = 3 x 10-4 N.m/(rad/s)

R = 1.2Ω

L = 0.020 H

Ke = 5 x 10-2 V/(rad/s)

Kt = 0.043 N.m/A

Substitute these parameters into the preceeding equations
toobtain the exact differential equations of the dc motor.
Determinethe transfer function, Ω(s)/E(s), by transforming all
threeequations into frequency-domain algebraic equations. Use
algebraicoperations to obtain the ratio of Ω/E, which is the
desiredtransfer function.

Here is what I have so far, then I got stuck:

Substitute the parameters into the 3 equations:

e = 12i + 0.020 di/dt + 5 x 10-2r/rad/sω

i = q/0.043 N.m/A

q = J/dω/dt

The method of finding the exact differential equation is notso
clear to me fromthe three equations. Should the 3 equations
beresolved into 1 differential equation? Please help with
astep-by-step method to solving these types of problems